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level3">满二叉树</a></li><!----><!--]--><!--[--><li class="toc-item"><a aria-current="page" href="/cs-basics/data-structure/tree.html#完全二叉树" class="router-link-active router-link-exact-active toc-link level3">完全二叉树</a></li><!----><!--]--><!--[--><li class="toc-item"><a aria-current="page" href="/cs-basics/data-structure/tree.html#平衡二叉树" class="router-link-active router-link-exact-active toc-link level3">平衡二叉树</a></li><!----><!--]--></ul><!--]--><!--[--><li class="toc-item"><a aria-current="page" href="/cs-basics/data-structure/tree.html#二叉树的存储" class="router-link-active router-link-exact-active toc-link level2">二叉树的存储</a></li><ul class="toc-list"><!--[--><li class="toc-item"><a aria-current="page" href="/cs-basics/data-structure/tree.html#链式存储" class="router-link-active router-link-exact-active toc-link level3">链式存储</a></li><!----><!--]--><!--[--><li class="toc-item"><a aria-current="page" href="/cs-basics/data-structure/tree.html#顺序存储" class="router-link-active router-link-exact-active toc-link level3">顺序存储</a></li><!----><!--]--></ul><!--]--><!--[--><li class="toc-item"><a aria-current="page" href="/cs-basics/data-structure/tree.html#二叉树的遍历" class="router-link-active router-link-exact-active toc-link level2">二叉树的遍历</a></li><ul class="toc-list"><!--[--><li class="toc-item"><a aria-current="page" href="/cs-basics/data-structure/tree.html#先序遍历" class="router-link-active router-link-exact-active toc-link level3">先序遍历</a></li><!----><!--]--><!--[--><li class="toc-item"><a aria-current="page" href="/cs-basics/data-structure/tree.html#中序遍历" class="router-link-active router-link-exact-active toc-link level3">中序遍历</a></li><!----><!--]--><!--[--><li class="toc-item"><a aria-current="page" href="/cs-basics/data-structure/tree.html#后序遍历" class="router-link-active router-link-exact-active toc-link level3">后序遍历</a></li><!----><!--]--></ul><!--]--></ul></div></aside></div><!----><div class="theme-hope-content"><!--[--><h1 id="树" tabindex="-1"><a class="header-anchor" href="#树" aria-hidden="true">#</a> 树</h1><p>树就是一种类似现实生活中的树的数据结构（倒置的树）。任何一颗非空树只有一个根节点。</p><p>一棵树具有以下特点：</p><ol><li>一棵树中的任意两个结点有且仅有唯一的一条路径连通。</li><li>一棵树如果有 n 个结点，那么它一定恰好有 n-1 条边。</li><li>一棵树不包含回路。</li></ol><p>下图就是一颗树，并且是一颗二叉树。</p><p><img src="https://my-blog-to-use.oss-cn-beijing.aliyuncs.com/2019-6/二叉树-2.png" alt="二叉树" loading="lazy"></p><p>如上图所示，通过上面这张图说明一下树中的常用概念：</p><ul><li><strong>节点</strong> ：树中的每个元素都可以统称为节点。</li><li><strong>根节点</strong> ：顶层节点或者说没有父节点的节点。上图中 A 节点就是根节点。</li><li><strong>父节点</strong> ：若一个节点含有子节点，则这个节点称为其子节点的父节点。上图中的 B 节点是 D 节点、E 节点的父节点。</li><li><strong>子节点</strong> ：一个节点含有的子树的根节点称为该节点的子节点。上图中 D 节点、E 节点是 B 节点的子节点。</li><li><strong>兄弟节点</strong> ：具有相同父节点的节点互称为兄弟节点。上图中 D 节点、E 节点的共同父节点是 B 节点，故 D 和 E 为兄弟节点。</li><li><strong>叶子节点</strong> ：没有子节点的节点。上图中的 D、F、H、I 都是叶子节点。</li><li><strong>节点的高度</strong> ：该节点到叶子节点的最长路径所包含的边数。</li><li><strong>节点的深度</strong> ：根节点到该节点的路径所包含的边数</li><li><strong>节点的层数</strong> ：节点的深度+1。</li><li><strong>树的高度</strong> ：根节点的高度。</li></ul><blockquote><p>关于树的深度和高度的定义可以看 stackoverflow 上的这个问题：<a href="https://stackoverflow.com/questions/2603692/what-is-the-difference-between-tree-depth-and-height" target="_blank" rel="noopener noreferrer">What is the difference between tree depth and height?<span><svg class="external-link-icon" xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewbox="0 0 100 100" width="15" height="15"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path><polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg><span class="external-link-icon-sr-only">open in new window</span></span></a> 。</p></blockquote><h2 id="二叉树的分类" tabindex="-1"><a class="header-anchor" href="#二叉树的分类" aria-hidden="true">#</a> 二叉树的分类</h2><p><strong>二叉树</strong>（Binary tree）是每个节点最多只有两个分支（即不存在分支度大于 2 的节点）的树结构。</p><p><strong>二叉树</strong> 的分支通常被称作“<strong>左子树</strong>”或“<strong>右子树</strong>”。并且，<strong>二叉树</strong> 的分支具有左右次序，不能随意颠倒。</p><p><strong>二叉树</strong> 的第 i 层至多拥有 <code>2^(i-1)</code> 个节点，深度为 k 的二叉树至多总共有 <code>2^(k+1)-1</code> 个节点（满二叉树的情况），至少有 2^(k) 个节点（关于节点的深度的定义国内争议比较多，我个人比较认可维基百科对<a href="https://zh.wikipedia.org/wiki/%E6%A0%91_(%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84)#/%E6%9C%AF%E8%AF%AD" target="_blank" rel="noopener noreferrer">节点深度的定义<span><svg class="external-link-icon" xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewbox="0 0 100 100" width="15" height="15"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path><polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg><span class="external-link-icon-sr-only">open in new window</span></span></a>）。</p><p><img src="https://guide-blog-images.oss-cn-shenzhen.aliyuncs.com/github/javaguide/image-20220119112736158.png" alt="" loading="lazy"></p><h3 id="满二叉树" tabindex="-1"><a class="header-anchor" href="#满二叉树" aria-hidden="true">#</a> 满二叉树</h3><p>一个二叉树，如果每一个层的结点数都达到最大值，则这个二叉树就是 <strong>满二叉树</strong>。也就是说，如果一个二叉树的层数为 K，且结点总数是(2^k) -1 ，则它就是 <strong>满二叉树</strong>。如下图所示：</p><p><img src="/assets/满二叉树.9025b079.png" alt="" loading="lazy"></p><h3 id="完全二叉树" tabindex="-1"><a class="header-anchor" href="#完全二叉树" aria-hidden="true">#</a> 完全二叉树</h3><p>除最后一层外，若其余层都是满的，并且最后一层或者是满的，或者是在右边缺少连续若干节点，则这个二叉树就是 <strong>完全二叉树</strong> 。</p><p>大家可以想象为一棵树从根结点开始扩展，扩展完左子节点才能开始扩展右子节点，每扩展完一层，才能继续扩展下一层。如下图所示：</p><p><img src="/assets/完全二叉树.c4f02f51.png" alt="" loading="lazy"></p><p>完全二叉树有一个很好的性质：<strong>父结点和子节点的序号有着对应关系。</strong></p><p>细心的小伙伴可能发现了，当根节点的值为 1 的情况下，若父结点的序号是 i，那么左子节点的序号就是 2i，右子节点的序号是 2i+1。这个性质使得完全二叉树利用数组存储时可以极大地节省空间，以及利用序号找到某个节点的父结点和子节点，后续二叉树的存储会详细介绍。</p><h3 id="平衡二叉树" tabindex="-1"><a class="header-anchor" href="#平衡二叉树" aria-hidden="true">#</a> 平衡二叉树</h3><p><strong>平衡二叉树</strong> 是一棵二叉排序树，且具有以下性质：</p><ol><li>可以是一棵空树</li><li>如果不是空树，它的左右两个子树的高度差的绝对值不超过 1，并且左右两个子树都是一棵平衡二叉树。</li></ol><p>平衡二叉树的常用实现方法有 <strong>红黑树</strong>、<strong>AVL 树</strong>、<strong>替罪羊树</strong>、<strong>加权平衡树</strong>、<strong>伸展树</strong> 等。</p><p>在给大家展示平衡二叉树之前，先给大家看一棵树：</p><p><img src="" alt="" loading="lazy"></p><p><strong>你管这玩意儿叫树？？？</strong></p><p>没错，这玩意儿还真叫树，只不过这棵树已经退化为一个链表了，我们管它叫 <strong>斜树</strong>。</p><p><strong>如果这样，那我为啥不直接用链表呢?</strong></p><p>谁说不是呢？</p><p>二叉树相比于链表，由于父子节点以及兄弟节点之间往往具有某种特殊的关系，这种关系使得我们在树中对数据进行<strong>搜索</strong>和<strong>修改</strong>时，相对于链表更加快捷便利。</p><p>但是，如果二叉树退化为一个链表了，那么那么树所具有的优秀性质就难以表现出来，效率也会大打折，为了避免这样的情况，我们希望每个做 “家长”（父结点） 的，都 <strong>一碗水端平</strong>，分给左儿子和分给右儿子的尽可能一样多，相差最多不超过一层，如下图所示：</p><p><img src="" alt="" loading="lazy"></p><h2 id="二叉树的存储" tabindex="-1"><a class="header-anchor" href="#二叉树的存储" aria-hidden="true">#</a> 二叉树的存储</h2><p>二叉树的存储主要分为 <strong>链式存储</strong> 和 <strong>顺序存储</strong> 两种：</p><h3 id="链式存储" tabindex="-1"><a class="header-anchor" href="#链式存储" aria-hidden="true">#</a> 链式存储</h3><p>和链表类似，二叉树的链式存储依靠指针将各个节点串联起来，不需要连续的存储空间。</p><p>每个节点包括三个属性：</p><ul><li>数据 data。data 不一定是单一的数据，根据不同情况，可以是多个具有不同类型的数据。</li><li>左节点指针 left</li><li>右节点指针 right。</li></ul><p>可是 JAVA 没有指针啊！</p><p>那就直接引用对象呗（别问我对象哪里找）</p><p><img src="/assets/链式存储二叉树.8dfe7691.png" alt="" loading="lazy"></p><h3 id="顺序存储" tabindex="-1"><a class="header-anchor" href="#顺序存储" aria-hidden="true">#</a> 顺序存储</h3><p>顺序存储就是利用数组进行存储，数组中的每一个位置仅存储节点的 data，不存储左右子节点的指针，子节点的索引通过数组下标完成。根结点的序号为 1，对于每个节点 Node，假设它存储在数组中下标为 i 的位置，那么它的左子节点就存储在 2i 的位置，它的右子节点存储在下标为 2i+1 的位置。</p><p>一棵完全二叉树的数组顺序存储如下图所示：</p><p><img src="/assets/顺序存储.ec1113ca.png" alt="" loading="lazy"></p><p>大家可以试着填写一下存储如下二叉树的数组，比较一下和完全二叉树的顺序存储有何区别：</p><p><img src="/assets/顺序存储2.941d44fb.png" alt="" loading="lazy"></p><p>可以看到，如果我们要存储的二叉树不是完全二叉树，在数组中就会出现空隙，导致内存利用率降低</p><h2 id="二叉树的遍历" tabindex="-1"><a class="header-anchor" href="#二叉树的遍历" aria-hidden="true">#</a> 二叉树的遍历</h2><h3 id="先序遍历" tabindex="-1"><a class="header-anchor" href="#先序遍历" aria-hidden="true">#</a> 先序遍历</h3><p><img src="/assets/先序遍历.05f522c6.png" alt="" loading="lazy"></p><p>二叉树的先序遍历，就是先输出根结点，再遍历左子树，最后遍历右子树，遍历左子树和右子树的时候，同样遵循先序遍历的规则，也就是说，我们可以递归实现先序遍历。</p><p>代码如下：</p><div class="language-java ext-java line-numbers-mode"><pre class="language-java"><code><span class="token keyword">public</span> <span class="token keyword">void</span> <span class="token function">preOrder</span><span class="token punctuation">(</span><span class="token class-name">TreeNode</span> root<span class="token punctuation">)</span><span class="token punctuation">{</span>
	<span class="token keyword">if</span><span class="token punctuation">(</span>root <span class="token operator">==</span> <span class="token keyword">null</span><span class="token punctuation">)</span><span class="token punctuation">{</span>
		<span class="token keyword">return</span><span class="token punctuation">;</span>
	<span class="token punctuation">}</span>
	system<span class="token punctuation">.</span>out<span class="token punctuation">.</span><span class="token function">println</span><span class="token punctuation">(</span>root<span class="token punctuation">.</span>data<span class="token punctuation">)</span><span class="token punctuation">;</span>
	<span class="token function">preOrder</span><span class="token punctuation">(</span>root<span class="token punctuation">.</span>left<span class="token punctuation">)</span><span class="token punctuation">;</span>
	<span class="token function">preOrder</span><span class="token punctuation">(</span>root<span class="token punctuation">.</span>right<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
</code></pre><div class="line-numbers" aria-hidden="true"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br><span class="line-number">4</span><br><span class="line-number">5</span><br><span class="line-number">6</span><br><span class="line-number">7</span><br><span class="line-number">8</span><br></div></div><h3 id="中序遍历" tabindex="-1"><a class="header-anchor" href="#中序遍历" aria-hidden="true">#</a> 中序遍历</h3><p><img src="/assets/中序遍历.bf6eff45.png" alt="" loading="lazy"></p><p>二叉树的中序遍历，就是先递归中序遍历左子树，再输出根结点的值，再递归中序遍历右子树，大家可以想象成一巴掌把树压扁，父结点被拍到了左子节点和右子节点的中间，如下图所示：</p><p><img src="" alt="" loading="lazy"></p><p>代码如下：</p><div class="language-java ext-java line-numbers-mode"><pre class="language-java"><code><span class="token keyword">public</span> <span class="token keyword">void</span> <span class="token function">inOrder</span><span class="token punctuation">(</span><span class="token class-name">TreeNode</span> root<span class="token punctuation">)</span><span class="token punctuation">{</span>
	<span class="token keyword">if</span><span class="token punctuation">(</span>root <span class="token operator">==</span> <span class="token keyword">null</span><span class="token punctuation">)</span><span class="token punctuation">{</span>
		<span class="token keyword">return</span><span class="token punctuation">;</span>
	<span class="token punctuation">}</span>
	<span class="token function">inOrder</span><span class="token punctuation">(</span>root<span class="token punctuation">.</span>left<span class="token punctuation">)</span><span class="token punctuation">;</span>
	system<span class="token punctuation">.</span>out<span class="token punctuation">.</span><span class="token function">println</span><span class="token punctuation">(</span>root<span class="token punctuation">.</span>data<span class="token punctuation">)</span><span class="token punctuation">;</span>
	<span class="token function">inOrder</span><span class="token punctuation">(</span>root<span class="token punctuation">.</span>right<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
</code></pre><div class="line-numbers" aria-hidden="true"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br><span class="line-number">4</span><br><span class="line-number">5</span><br><span class="line-number">6</span><br><span class="line-number">7</span><br><span class="line-number">8</span><br></div></div><h3 id="后序遍历" tabindex="-1"><a class="header-anchor" href="#后序遍历" aria-hidden="true">#</a> 后序遍历</h3><p><img src="/assets/后序遍历.7b5edc1b.png" alt="" loading="lazy"></p><p>二叉树的后序遍历，就是先递归后序遍历左子树，再递归后序遍历右子树，最后输出根结点的值</p><p>代码如下：</p><div class="language-java ext-java line-numbers-mode"><pre class="language-java"><code><span class="token keyword">public</span> <span class="token keyword">void</span> <span class="token function">postOrder</span><span class="token punctuation">(</span><span class="token class-name">TreeNode</span> root<span class="token punctuation">)</span><span class="token punctuation">{</span>
	<span class="token keyword">if</span><span class="token punctuation">(</span>root <span class="token operator">==</span> <span class="token keyword">null</span><span class="token punctuation">)</span><span class="token punctuation">{</span>
		<span class="token keyword">return</span><span class="token punctuation">;</span>
	<span class="token punctuation">}</span>
	<span class="token function">postOrder</span><span class="token punctuation">(</span>root<span class="token punctuation">.</span>left<span class="token punctuation">)</span><span class="token punctuation">;</span>
	<span class="token function">postOrder</span><span class="token punctuation">(</span>root<span class="token punctuation">.</span>right<span class="token punctuation">)</span><span class="token punctuation">;</span>
	system<span class="token punctuation">.</span>out<span class="token punctuation">.</span><span class="token function">println</span><span class="token punctuation">(</span>root<span class="token punctuation">.</span>data<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
</code></pre><div class="line-numbers" aria-hidden="true"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br><span class="line-number">4</span><br><span class="line-number">5</span><br><span class="line-number">6</span><br><span class="line-number">7</span><br><span class="line-number">8</span><br></div></div><!--]--></div><!----><footer class="page-meta"><div class="meta-item edit-link"><a href="https://github.com/Snailclimb/JavaGuide/edit/main/docs/cs-basics/data-structure/tree.md" rel="noopener noreferrer" target="_blank" arialabel="编辑此页" class="nav-link label"><!--[--><svg xmlns="http://www.w3.org/2000/svg" class="icon edit-icon" viewbox="0 0 1024 1024" arialabelledby="edit"><title id="edit" lang="en">edit icon</title><g fill="currentColor"><path d="M430.818 653.65a60.46 60.46 0 0 1-50.96-93.281l71.69-114.012 7.773-10.365L816.038 80.138A60.46 60.46 0 0 1 859.225 62a60.46 60.46 0 0 1 43.186 18.138l43.186 43.186a60.46 60.46 0 0 1 0 86.373L588.879 565.55l-8.637 8.637-117.466 68.234a60.46 60.46 0 0 1-31.958 11.229z"></path><path d="M728.802 962H252.891A190.883 190.883 0 0 1 62.008 771.98V296.934a190.883 190.883 0 0 1 190.883-192.61h267.754a60.46 60.46 0 0 1 0 120.92H252.891a69.962 69.962 0 0 0-69.098 69.099V771.98a69.962 69.962 0 0 0 69.098 69.098h475.911A69.962 69.962 0 0 0 797.9 771.98V503.363a60.46 60.46 0 1 1 120.922 0V771.98A190.883 190.883 0 0 1 728.802 962z"></path></g></svg><!--]-->编辑此页<span><svg class="external-link-icon" xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewbox="0 0 100 100" width="15" height="15"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path><polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg><span class="external-link-icon-sr-only">open in new window</span></span><!----></a></div><div class="meta-item update-time"><span class="label">上次编辑于: </span><span class="info">2022/4/7 11:34:43</span></div><div class="meta-item contributors"><span class="label">贡献者: </span><!--[--><!--[--><span class="contributor" title="email: zxy690@qq.com">Dayu</span>,<!--]--><!--[--><span class="contributor" title="email: koushuangbwcx@163.com">guide</span><!--]--><!--]--></div></footer><nav class="page-nav"><a href="/cs-basics/data-structure/heap.html" class="nav-link prev" arialabel="堆"><div class="hint"><span class="arrow left"></span>上一页</div><div class="link"><!---->堆</div></a><a href="/cs-basics/data-structure/red-black-tree.html" class="nav-link next" arialabel="红黑树"><div class="hint">下一页<span class="arrow right"></span></div><div class="link">红黑树<!----></div></a></nav><!----><!----></main><!--]--><footer class="footer-wrapper"><div class="footer"><a href="https://beian.miit.gov.cn/" target="_blank">鄂ICP备2020015769号-1</a></div><div class="copyright">Copyright © 2022 Guide</div></footer></div><!--]--><!----><!--]--></div>
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